Thank you, Keith-san!It's used in conversation, but usually refers to an activity rather than, say, a movie. E.g:
Late-night parties are all the rage.
Drinking tequila is all the rage.
Taking selfies on your mobile phone is all the rage...
I don't think it's so much that the expression is dated, rather that it is usually used to describe something that was popular at some time in the past. For example, miniskirts for women were all the rage in the sixties; long hair for men was all the rage in the seventies.All the rage sounds very dated to me - from the 1950s perhaps.